Extended Data Table 1. Design parameters for Fourier surfaces.

Design parameters for all Fourier surfaces demonstrated in this work. The functions are defined for the design to be patterned in the polymer surface, where x and y lie in plane and z is perpendicular (pointing away from the substrate). A right-handed coordinate system is used with the origin placed in the middle of the pattern in both the x and y directions. In these formulas, the height of the surface is defined relative to the unpatterned flat surface where z = 0. All A_t and Δ_i (𝛬_i) have been rounded to the nearest 0.1 (1.0) nm. Analysis of the measured topographies for templated Ag gratings shows that the 𝛬_i are consistently ~2% larger than the design value (Extended Data Fig. 2), attributed to a distance miscalibration in the thermal scanning probe. See Methods.

	Parameters										Height profile
Figure	A 1 (nm)	A 2 (nm)	A 3 (nm)	𝛬 1 (nm)	𝛬 2 (nm)	𝛬 3 (nm)	φ 1 (nm)	φ 2 (nm)	φ 3 (nm)	Δ (nm)	f(x)
Fig. 1a	25.0	–	–	600	–	–	180	–	–	35.0	$\sum _i{A}_i\cos (g_{i}x+{\phi }_i)-\Delta$
Fig. 1d	18.1	14.3	–	600	475	–	0	0	–	42.3
Fig. 1g	18.4	7.0	6.4	600	230	210	0	0	0	41.8
ED Fig. 5	19.3	9.6	–	620	310	–	180	180	–	24.4
								0		38.9
ED Fig. 6	18.5	0 to 25.1	–	620	230	–	0	0	–	28.0 to 53.1
Fig. 4b	12.0	12.0	12.0	620	520	445	0	0	0	45.9
ED Fig.8a,e	100	-	-	1000	-	-	180	-	-	110
ED Fig.8c	88.6	44.3	-	800	400	-	90	90	-	125
ED Fig.8g	88.6	44.3	-	900	450	-	90	90	-	125
ED Fig.9b	25.0	–	–	400	–	–	180	–	–	35.0
ED Fig.9d	14.1	11.8	10.1	414	347	297	0	0	0	45.8
Figure	i	Ai (nm)	𝛬i (nm)	φi (deg)	θ 1,7 (deg)	θ 2,8 (deg)	θ 3,9 (deg)	θ 4,10 (deg)	θ 5,11 (deg)	θ 6,12 (deg)	Δ (nm)	f(x,y)
Fig. 2a	1,2	17.5	600	0	0	-10	–	–	–	–	45.0	$\sum _i{A}_i\cos [g_i(x\cos {\theta }_i+y\sin {\theta }_i)+{\phi }_i]-\Delta$
Fig. 2b	1,2	17.5	600	0	0	-40	–	–	–	–	45.0
Fig. 3a	1,2,3	15.6	600	0	0	60	120	–	–	–	56.7
Fig. 3d	1-6	10.0	600	0	0	30	60	90	120	150	70.0
Fig. 4e	1–6	5.6	700	0	0	30	60	90	120	150	77.3
	7–12	5.6	308	0	0	30	60	90	120	150	77.3
ED Fig. 9e,f	1-6	11.1	615	0	0	30	60	90	120	150	76.7
Figure	i	A 1 (nm)	A 2 (nm)	𝛬 1 (nm)	𝛬 2 (nm)	φ 1 (deg)	φ 2 (deg)	Δ (nm)	r1 (nm)	r2 (nm)	f(r,θ)
ED Fig. 7c	1	35.0	–	600	–	180	–	45	0	–	$\sum _i{A}_i\cos (g_i|r-r_i|+{\phi }_i)-\Delta$
ED Fig. 7d	1,2	17.5	17.5	600	600	180	0	45	–150 ŷ	150 ŷ
Figure	A (nm)	L (nm)		Δ (nm)	f(r)
Fig. 2e	35.0	1581		45.0	$A\sin [\pi {(\frac{r}{L})}^2]-\Delta$